Electric Effects of Insulator Layer in Presence of Multipathing

Electron multipathing in the presence of an insulator layer G. Iadarola, M. Taborelli Many thanks to L. Giacomel, E. Metral, V. Petit, G. Rumolo

Outlook • Introduction • Electric effects • Surface effects • Simulation tests • Two insulator patches

Introduction • Stains been observed on spare LHC beam screens • When attempting SEY measurements it was found that some of the stains are charging  they behave like an insulating layer • Insulators typically have high SEY, but their SEY depends on the charge state • What is the impact on the e-cloud buildup? • Our test scenario consists in a copper chamber with a single attached insulating patch • Caveat: • As we have no quantitative information on the behavior of these spots, it is not possible to make any quantitative estimate • We will instead try to explore possible mechanisms and behaviors. Insulator Copper

Conductor vs insulator • When secondary emission takes place (emission of more electrons than impacting ones): • A conductor remains neutral (can draw charges from the ground) • An insulator charges positively. This has two consequences: • Electric effect: charge on the surface can generate a field in the chamber, potentially changing the dynamics of the cloud • Surface effect: the behavior of the surface, in particular its SEY, change as a function of the charge state(1) - - - - - - Insulator + Conductor (1) NB: this has nothing to with usual conditioning (which is a “chemical” change), this is a “physical” change, which reverses when the surface discharges M. Belhaj, “SEY properties of dielectric materials, modelling and measurements”, ECLOUD18 workshop

Electric effect • In general, a charge distribution will generate an electric field in the beam chamber • When the insulator lies on a conducting substrate, charges are induced in the conductor which tend to cancel the field of the charge in the insulator PEC Charged patch Insulator Conductor PyPIC simulation + + + - - - • We expect the field in the chamber to become smaller when the insulator is thinner • We quantify this using a simple model…

Electric effect: some analytic estimate We consider a 1D simplification of the problem: x PEC x = h x Copper Vacuum (e0) Insulator Q = charge density on the surface of the insulator x = d Insulator (ei) x = 0 PEC

Electric effect: some analytic estimate We consider a 1D simplification of the problem: x We introduce the electrostatic potential: PEC x = h (1)  From Gauss’s law we can write Laplace’s equation: (2)  The boundary conditions are simply: Vacuum (e0) (3) Combining (2) and (3) the potential must be in the form: Q = charge density on the surface of the insulator in the insulator x = d Insulator (ei) in the vacuum x = 0 PEC

Electric effect: some analytic estimate We consider a 1D simplification of the problem: x x We introduce the electrostatic potential: PEC x = h (1)  From Gauss’s law we can write Laplace’s equation: (2)  The boundary conditions are simply: Vacuum (e0) (3) Combining (2) and (3) the potential must be in the form: Q in the insulator x = d Insulator (ei) in the vacuum x = 0 fi PEC f(x)

Electric effect: some analytic estimate We consider a 1D simplification of the problem: x x The potential must be in the form: PEC in the insulator x = h in the vacuum We impose the continuity of the potential:  Vacuum (e0) We are left with one unknown (i.e. a). Applying Gauss’s law at the charged surface:  Q x = d Insulator (ei) x = 0 fi PEC f(x) which gives:

Electric effect: some analytic estimate We consider a 1D simplification of the problem: x x We can write the potential explicitly: PEC x = h in the insulator in the vacuum From any of the two we can get the maximum potential, which occurs at the interface: Vacuum (e0) Q x = d Insulator (ei) As expected it becomes zero when d or Q tend to zaro. x = 0 fi PEC f(x)

Electric effect: some analytic estimate We consider a 1D simplification of the problem: x x Potential at the interface: PEC x = h For d << h, we simply get: Vacuum (e0) • The potential on the surface only depends on on the insulator thickness and material • It does not depend on the on the geometry of the “chamber” (h) • It is linearly proportional to the insulator thickness It vanishes for infinitely thin layer (as guessed by image charges considerations) • In reduces if the insulator has a high permittivity Q x = d Insulator (ei) x = 0 fi PEC f(x)

Electric effect: comparison against 2D Poisson solver • Even in the 2D geometry, the formula gives a very good approximation of the potential at the surface (potential in the rest of the chamber is instead overestimated) Patch width: 3 mm Patch width: 3 mm On the axis of the patch PyPIC simulation

Electric effect: comparison against 2D Poisson solver • Even in the 2D geometry, the formula gives a very good approximation of the potential at the surface (potential in the rest of the chamber is instead overestimated) Patch width: 10 mm Patch width: 10 mm On the axis of the patch PyPIC simulation • As expected, the approximation in the rest of the chamber gets betted for a wider patch

Electric effect: comparison against 2D Poisson solver • Even in the 2D geometry, the formula gives a very good approximation of the potential at the surface • Agreement becomes better for smaller thickness of the insulator • For realistic values of the insulator thickness, the potential is relatively small  in first approximation we will neglect the electrostatic effect of the charge on the patch Patch width: 3 mm Patch width: 10 mm

Surface effects • When an insulator emits electrons its valence band starts being depopulated  formation of holes • This affects the Secondary Electron Yield: • When the surface charges, the Secondary Electron Yield tends to 1.0 over a wide range of energies • This is a reversible process, the SEY recovers its initial value when the surface discharges Increasing charge M. Belhaj, “SEY properties of dielectric materials, modelling and measurements”, ECLOUD18 workshop

Surface effects: Secondary Electron Yield • An insulator module has been included in PyECLOUD: the code keeps track of the accumulated charge and adapts the SEY curve accordingly: • The “starting curve” (Q=0) uses the usual SEY models (custom SEYmax) • The “arrival curve” (Q >= Qmax) the SEY has the form (Qmaxand EQ are defined by the user): • For 0<Q< Qmaxa linear weighting between the two is used When charged the surface absorbs only low-energy electrons EQ

Surface effects: relaxation constant If the resistivity of the insulator is not infinite there will be a small current to ground Ohm’s law: Insulator (ei,ri) + + + + Ji From our previous calculation: Conductor - - - - Combining the two: which gives: with: • Even for relatively high resistivity the discharge can be quite fast: • Ex.ri = 107 W m  ti = 100 ms

Surface effects: implemented model • The charging module is built on top of the existing non-uniform SEY module • Can be activated by selecting switch_model = 'ECLOUD_nunif_charging’ • Surface properties can be defined independently for each segment of the chamber (via the chamber mat files) • Attributes are defined for all segments: • flag_charging decides which segments behave like insulators • Q_max_segments defines the charge density for which dmax is 1 • EQ_segments defines the shape of the SEY curve of the charged surface • tau_segments defines the charge relaxation time flag_charging = True flag_charging = False

Simulation tests: no discharging, low EQ dCu: 1.3 di(Q=0): 1.9 Qmax: 1.0e-13 C/mm2 EQ: 0.01 eV ti: Inf us Normal side Patch side For first tests we choose relatively small Qmax to see the effect over a short simulation Patch

Simulation tests: no discharging, low EQ dCu: 1.3 di(Q=0): 1.9 Qmax: 1.0e-13 C/mm2 EQ: 0.01 eV ti: Inf us Normal side Patch side The surface has practically no way of discharging Patch

Simulation tests: no discharging, low EQ dCu: 1.3 di(Q=0): 1.9 Qmax: 1.0e-13 C/mm2 EQ: 0.01 eV ti: Inf us Normal side Patch side At the beginning, the SEY is very high, the cloud ion the patch region builds up very quickly Then the surface charges up and the SEY of the patch rapidly drops to 1.0 Patch

Simulation tests: no discharging, higher EQ dCu: 1.3 di(Q=0): 1.9 Qmax: 1.0e-13 C/mm2 EQ: 20.0 eV ti: Inf us Normal side Patch side Patch

Simulation tests: no discharging, higher EQ dCu: 1.3 di(Q=0): 1.9 Qmax: 1.0e-13 C/mm2 EQ: 20.0 eV ti: Inf us Normal side Patch side The surface can discharge by absorbing low-energy electrons. The surface charges only up to a certain equilibrium level corresponding to a certain SEY Patch

Simulation tests: discharging dCu: 1.3 di(Q=0): 1.9 Qmax: 1.0e-13 C/mm2 EQ: 20.0 eV ti: 2.0 us We introduce a discharging time-constant Normal side Patch side The equilibrium charge decreases and the the equilibrium SEY increases Patch

A “realistic” simulation • From lab measurements on insulators we know that Qmax = ~10-10 C/mm2 • ri = 107 W m  ti = 100 ms • EQ = 20 eV

Simulation tests: a realistic simulation dCu: 1.3 di(Q=0): 1.9 Qmax: 1.0e-10 C/mm2 EQ: 20.0 eV ti: 100 us Normal side Patch side We need to simulate three LHC turns Patch Due to relatively quick discharge and large Qmaxthe equilibrium level is very low The equilibrium SEY is very large

A crosscheck: two patches facing each other • At some point the electrons in the column will be limited by their space charge • This will limit also the charge on the patches and therefore the decrease in SEY Conductor Insulator + + - B This is a closed system: Qpatches = - Qele (subtracting seed electrons) - - - Insulator + + Conductor

Simulation tests: two patches facing each other dCu: 1.3 di(Q=0): 1.9 Qmax: 1.0e-13 C/mm2 EQ: 20 eV ti: Inf us Qpatches = - Qeleisindeed verified flag_charging = True flag_charging = False |charge on the patches| |charge e-cloud| Charge saturates when e-cloud does Only insulator area is seeded (no e-cloud elsewhere) Equilibrium SEY remains high Patch

Summary • In the presence of an insulating layer on a beam pipe, charge can accumulate on the surface • If the layer is sufficiently thin, there is no significant field induced in the pipe (charge induced in the conductor behind) • Experiments show that the accumulation of charge affects also the Secondary Electron Yield, in particular it pushes it towards 1.0 • The surface can discharge due to different mechanisms. Two effects were considered here • Absorption of low-energy electrons (SEY < 1.0 at at very low energies) • Conductivity is poor but not zero • PyECLOUD has been extended to include these mechanisms and investigate the dynamics • Simulations show that an equilibrium charge is found as a result of a balance between charging and discharging mechanisms •  this results in an equilibrium SEY on the patch surface • For some plausible numbers (di = 1.9, Qmax=1.0e-10 C/mm2, ti: 100 us), due to a relatively fast discharging, the SEY can remain quite high with a visible effect on the heat loads • For quantitative estimates a lab characterization of the insulator is needed… stay tuned…

Electric Effects of Insulator Layer in Presence of Multipathing

Electric Effects of Insulator Layer in Presence of Multipathing

Presentation Transcript

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